LBan-1

Overview

This lesson is structured in three parts:

• Self-averaging and disorder in statistical systems

Disordered systems are characterized by a random energy landscape, however, in the thermodynamic limit, physical observables become deterministic. This property, known as self-averaging, does not always hold for the partition function which is the quantity that we can compute. When it holds the annealed average $\ln {\overline {Z}}$ and the quenched average $\ln Z$ coincides otherwiese we have

$\ln Z<\ln {\overline {Z}}$

• The Random Energy Model

We study the Random Energy Model (REM) introduced by Bernard Derrida. In this model at each configuration is assigned an independent energy drawn from a Gaussian distribution. The model exhibits a freezing transition at a critical temperature​, below which the free energy becomes dominated by the lowest energy states.

• Extreme value statistics and saddle-point analysis

The results obtained from a saddle-point approximation can be recovered using the tools of extreme value statistics. In the REM, the low-temperature phase is governed by the minimum of a large set of independent energy values.

Part I

Random energy landascape

In a system with $N$ degrees of freedom, the number of configurations grows exponentially with $N$. For simplicity, consider Ising spins that take two values, $\sigma _{i}=\pm 1$, located on a lattice of size $L$ in $d$ dimensions. In this case, $N=L^{d}$ and the number of configurations is $M=2^{N}=e^{N\log 2}$.

In the presence of disorder, the energy associated with a given configuration becomes a random quantity. For instance, in the Edwards-Anderson model:

$E=-\sum _{\langle i,j\rangle }J_{ij}\sigma _{i}\sigma _{j},$

where the sum runs over nearest neighbors $\langle i,j\rangle$, and the couplings $J_{ij}$ are independent and identically distributed (i.i.d.) Gaussian random variables with zero mean and unit variance.

The energy of a given configuration is a random quantity because each system corresponds to a different realization of the disorder. In an experiment, this means that each of us has a different physical sample; in a numerical simulation, it means that each of us has generated a different set of couplings $J_{ij}$.

To illustrate this, consider a single configuration, for example the one where all spins are up. The energy of this configuration is given by the sum of all the couplings between neighboring spins:

$E[\sigma _{1}=1,\sigma _{2}=1,\ldots ]=-\sum _{\langle i,j\rangle }J_{ij}.$

Since the the couplings are random, the energy associated with this particular configuration is itself a Gaussian random variable, with zero mean and a variance proportional to the number of terms in the sum — that is, of order $N$. The same reasoning applies to each of the $M=2^{N}$ configurations. So, in a disordered system, the entire energy landscape is random and sample-dependent.

Self-averaging observables

A crucial question is whether the physical properties measured on a given sample are themselves random or not. Our everyday experience suggests that they are not: materials like glass, ceramics, or bronze have well-defined, reproducible physical properties that can be reliably controlled for industrial applications.

From a more mathematical point of view, it means tha physical observables — such as the free energy $F_{N}(\beta )=Nf_{N}(\beta )$ and its derivatives (magnetization, specific heat, susceptibility, etc.) — are self-averaging. This means that, in the limit $N\to \infty$, the distribution of the observable concentrates around its average:

$\lim _{N\to \infty }f_{N}=\lim _{N\to \infty }f_{N}^{\text{typ}}=\lim _{N\to \infty }{\overline {f_{N}}},\quad \quad {\overline {f_{N}}}=\int \,df\,P_{f_{N}}(f)\,f$

Hence macroscopic observables become effectively deterministic and their fluctuations from sample to sample vanish in relative terms:

$\lim _{N\to \infty }{\frac {\overline {f_{N}^{2}}}{{\overline {f_{N}}}^{2}}}=1.$

The partition function

The partition function

$Z_{N}=\exp(-\beta Nf_{N})$

is itself a random variable in disordered systems. What we are interested in computing is the quenched average of the free energy: $f_{N}\sim {\overline {f_{N}}}=-\ln Z_{N}/(\beta N)$ However, what is usually easier to compute is the annealed average: $-{\frac {1}{\beta N}}\ln \,{\overline {Z_{N}}}$

Do these two averages coincide?

If the partition function is self-averaging in the thermodynamic limit, then ${\overline {Z_{N}}}\equiv Z_{N}^{\text{typ}}$, and as a consequence, the annealed and quenched averages coincide.

However, the partition function is not always self-averaging. This happens when extremely rare configurations contribute disproportionately to its moments. In this case, the annealed average overestimates the true typical behavior of the system.

There are then two main strategies:

• One can directly compute the quenched average using methods such as the Parisi solution.

• Alternatively, one may determine the typical value $Z_{N}^{\text{typ}}$ and evaluate $f_{N}^{\text{typ}}=-{\frac {1}{\beta N}}\ln Z_{N}^{\text{typ}}$ in the limit $N\to \infty$.

Part II

Random Energy Model

This model simplifies the problem by neglecting correlations between the $M=2^{N}$ configurations and assuming that the energies $E_{\alpha }$ are(i.i.d.) Gaussian variables with zero mean and variance $E_{N}$.

Number of states is given by:

$p(E_{\alpha })={\frac {1}{\sqrt {2\pi N}}}\exp \left(-{\frac {E_{\alpha }^{2}}{2N}}\right)$